Wavelength reuse for efficient packet-switched transport

JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 21, NO. 6, JUNE 2003 1435

Wavelength Reuse for Efficient Packet-SwitchedTransport in an AWG-Based Metro WDM NetworkMichael Scheutzow, Martin Maier, Martin Reisslein, Member, IEEE, and Adam Wolisz, Senior Member, IEEE

Abstract—Metro wavelength-division multiplexed (WDM) net-works play an important role in the emerging Internet hierarchy;they interconnect the backbone WDM networks and the local-ac-cess networks. The current circuit-switched SONET/synchronousdigital hierarchy (SDH)-over-WDM-ring metro networks areexpected to become a serious bottleneck—the so-called metrogap—as they are faced with an increasing amount of bursty packetdata traffic and quickly increasing bandwidths in the backbonenetworks and access networks. Innovative metro WDM networksthat are highly efficient and able to handle variable-size packetsare needed to alleviate the metro gap. In this paper, we studyan arrayed-waveguide grating (AWG)-based single-hop WDMmetro network. We analyze the photonic switching of variable-sizepackets with spatial wavelength reuse. We derive computationallyefficient and accurate expressions for the network throughputand delay. Our extensive numerical investigations—based on ouranalytical results and simulations—reveal that spatial wavelengthreuse is crucial for efficient photonic packet switching. In typicalscenarios, spatial wavelength reuse increases the throughput by60% while reducing the delay by 40%. Also, the throughput ofour AWG-based network with spatial wavelength reuse is roughly70% larger than the throughput of a comparable single-hopWDM network based on a passive star coupler (PSC).

Index Terms—Arrayed-waveguide grating (AWG), medium ac-cess control, metro wavelength-division multiplexed (WDM) net-work, multiple free spectral ranges, passive star coupler (PSC),photonic packet switching, spatial wavelength reuse.

I. INTRODUCTION

T HE Internet of the future may be viewed as a three-levelhierarchy consisting of backbone networks, metro net-

works, and access networks. Backbone networks will providealmost infinite bandwidth based on wavelength-division multi-plexing (WDM) links. These WDM links are connected withreconfigurable all-optical add–drop multiplexers (OADMs) andall-optical cross connects (OXCs) controlled by multiprotocollambda switching (MPS) [1], optical burst switching (OBS)[2], and optical packet switching (OPS) [3]–[5] mechanisms.Access networks transport data to (and from) individual users.

Manuscript received June 13, 2002; revised March 10, 2003. This work wassupported in part by the Federal German Ministry of Education and Researchwithin the TransiNet Project, the DFG Research Center “Mathematics for keytechnologies” (FZT86) Berlin, and by the National Science Foundation underGrant CAREER ANI–0133252. A shorter version of this paper has appeared inProc. IEEE INFOCOM, New York, June 2002, pp. 1432–1441 [47].

M. Scheutzow is with the Department of Mathematics, Technical Universityof Berlin, 10587 Berlin, Germany (e-mail: [email protected]).

M. Maier and A. Wolisz are with the Telecommunication Networks Group,Technical University of Berlin, 10587 Berlin, Germany (e-mail: [email protected]; [email protected]).

M. Reisslein is with the Department of Electrical Engineering, Arizona StateUniversity, Tempe, AZ 85287-7206 USA (e-mail: [email protected]).

Digital Object Identifier 10.1109/JLT.2003.812723

By employing advanced local-area network (LAN) technolo-gies, such as Gigabit Ethernet, broad-band access, such asxDSL and cable modems, as well as high-speed next-generationwireless systems, such as UMTS, access networks providean ever increasing amount of bandwidth. Metro networksinterconnect the high-speed WDM backbone networks andthe high-speed access networks. Current metro networks aretypically SONET/synchronous digital hierarchy (SDH)-over-WDM rings which are based on circuit switching and carry theever-increasing amount of bursty data traffic only inefficiently.In addition, content providers increasingly place proxy cachesin metro networks. These proxies further increase the loadon metro networks. Metro networks are therefore expected tobecome a serious bottleneck—the so-called metro gap—in thefuture Internet. For these reasons, there is an urgent need forinnovative metro network architectures and protocols [6].

Two key requirements for metro networks are 1) flexibility,and 2) efficiency. Flexibility is required since metro networkshave to support a wide range of heterogeneous protocols,such as asynchronous transfer mode (ATM), Frame Relay,SONET/SDH, and Internet protocol (IP). This requires, inparticular, that the metro networks are able to transport packetsof different sizes. Efficiency is required because metro net-works are highly cost sensitive. Therefore, the deployed WDMnetworking components and the WDM networking resources(in particular, wavelengths) must be utilized efficiently. As wedemonstrate in this paper, a crucial technique for achievinghigh efficiency isspatial wavelength reuse. By spatial wave-length reuse we mean that in our arrayed-waveguide grating(AWG)-based metro WDM network (outlined in Section II) allwavelengths are used at all AWG ports simultaneously.

This paper builds on earlier work [7], in which we have pro-posed a novel AWG-based single-hop WDM network, whichprovides a dramatically increased degree of concurrency by i)using multiple free spectral ranges (FSRs), ii) spatially reusingall wavelengths at each AWG port, and iii) exploiting spreadingtechniques to enable simultaneous transmission of control anddata. This earlier work focused primarily on the network archi-tecture and the medium access control (MAC) protocol. The el-ementary analysis conducted in [7] provided very limited in-sights into the performance of the proposed network. The per-formance analysis in [7] is limited in that it considered onlyfixed-size packets and did not consider spatial wavelength reuse.However, the efficient transmission of variable-size packets is ofparamount importance for future metro networks. In this paperwe study 1) the photonic switching of packets of different sizes,and 2) the spatial wavelength reuse in the AWG-based networkproposed in earlier work. The main contribution of this paperis to develop a stochastic model to evaluate the performance

0733-8724/03$17.00 © 2003 IEEE

1436 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 21, NO. 6, JUNE 2003

of photonic packet switching with spatial wavelength reuse inthe AWG-based network. Our analytical model gives compu-tationally efficient and accurate expressions for the throughputand delay in the network. Our numerical results indicate thatspatial wavelength reuse is crucial for efficient photonic packetswitching. For typical scenarios, spatial wavelength reuse in-creases the throughput by 60% while reducing the delay by 40%.

This paper is organized as follows. In the following subsec-tion, we give a quick overview of related work. In Section II,we briefly review the architecture of the studied AWG-basedmetro network as well as the reasoning for selecting this ar-chitecture. In Section III, we briefly review the MAC protocolfor the studied network. In Section IV, we develop a stochasticmodel for the performance evaluation of the transmission ofvariable-size packets with spatial wavelength reuse. This modeland performance evaluation are our main contributions. In Sec-tion V, we use our analytical results to conduct numerical inves-tigations. We also conduct simulations to verify the accuracy ofour analytical results. In Section VI, we report on additional ex-tensive simulations that i) examine the backoff in our MAC pro-tocol, ii) evaluate the network performance for larger schedulingwindows and nodal buffers, and iii) compare the throughput per-formance of our AWG-based network with that of a passive starcoupler (PSC)-based network. We conclude the paper in Sec-tion VII.

A. Related Work

Single-hop metro WDM networks based on the PSC havebeen studied extensively, see, for instance, [8]–[18] as wellas the surveys [19], [20]. The studied networks are so-calledbroadcast-and-selectnetworks. Each transmission is broadcastover the PSC to all nodes. The intended recipient processes thetransmission, whereas the other nodes ignore the transmission.The primary limitation of the PSC-based networks is that eachwavelength provides only one communication channel betweenany pair of network nodes at any point in time. Wavelengths,however, are scarce, especially with the low-cost coarse WDMtechnology that is used in cost-sensitive metro networks. Bypartitioning the network into several PSC-based clusters andinterconnecting the clusters in some fashion, the same set ofwavelengths can be reused at each cluster PSC [21]–[23].

The network studied in this paper is fundamentally differentfrom the PSC-based networks in that it exploits the wave-length-routing property of the AWG. In the studied AWG-basednetwork, transmissions are not broadcast. Instead, they areselectively forwarded by using the appropriate wavelength, asdiscussed in detail later. This results in aswitchednetworkas opposed to the aforementioned broadcast networks. Inaddition, in our network, wavelengths are spatially reused atthe different AWG input ports, which allows for a high degreeof concurrency and efficient use of the scarce wavelengthresources.

Apart from a few older studies, such as [8], [24], [25], metroWDM networks have just recently begun to attract the interestfrom the research community [26]. A metro network basedon OADMs has recently been studied in [27]. This network isgeared toward opticalcircuit switching. In [28], a ring network

Fig. 1. Wavelength routing in AWG.

employing dynamic wavelength add–drop multiplexers isstudied. Slotted packet-switched ring networks are studied in[29], [30]. In [31], a ring WDM network with AWG basedOADMs has been proposed whose frequency-cyclic nature canbe used for easily upgrading the network capacity. For inter-connecting such WDM rings in an efficient and cost-effectiveway, the AWG was used as a passive wavelength-routing hub in[32]. General design principles and architectures for networksbased on AWGs are studied in [33]–[38]. On-line schedulingalgorithms for an AWG-based network with static wavelengthassignment and static time-division multiple access, which isfundamentally different from the on-demand reservations inour network, are studied in [39]. A protection routing strategyemploying AWGs is developed in [40] and a packet switchbased on an AWG is studied in [41].

The HORNET metro network [42], [43] allows for opticalpacketswitching. HORNET [42], [43] and the metro networksstudied in [27]–[31] have a ring topology, i.e., transmissionstypically have to traverse multiple network nodes. These net-works are therefore fundamentally different from the packet-switched single-hop network studied here. We note that AWG-based single-hop WDM networks are also studied in [44]–[46].However, these networks have either a more complex hub struc-ture with wavelength converters and a centralized resource man-agement [44], [45] or require a large number of transceiversat each node [46]. In contrast, our network is completelypas-siveand resources are allocated in adistributedfashion (as de-scribed in the subsequent sections). Moreover,multipleFSRs ofthe underlying AWG are used to increase the degree of concur-rency and thereby to improve the throughput-delay performancesignificantly. Each node is equipped with onesingle tunabletransceiver and an off-the-shelf broad-band light source [e.g.,light-emitting diode (LED)]. Finally, the novel node architectureof the proposed metro network allows for simultaneous trans-mission of data and controlwithoutrequiring a separate controlchannel or an additional receiver at each node. Therefore, allwavelengths can be used for data transmission.

We note that a summary of the results of this paper has ap-peared in a shorter conference paper [47]. The conference paperstates only the final analytical result of the throughput-delayanalysis and provides numerical results on the impact of a subsetof the system parameters. In contrast, in this extended paper,we present the derivation of the analytical results and examinethe impact of all system parameters numerically. Furthermore,this extended paper provides additional simulation results on theMAC protocol backoff, the impact of scheduling window andnodal buffers, as well as a comparison with a PSC-based net-work.

SCHEUTZOWet al.: WAVELENGTH REUSE FOR EFFICIENT PACKET-SWITCHED TRANSPORT 1437

Fig. 2. Network and node architecture.

II. A RCHITECTURE

In this section, we outline the architecture of the studied AWGbased network. First, we briefly review the physical proper-ties of the AWG. The AWG [48]–[50] is a passive wavelength-routing device with a wide range of applications [51], [52]. Inour network, we use the AWG as wavelength router [53]. Wenote that the recent development of AWGs with a crosstalk aslow as 40 dB makes it possible to spatially reuse all wave-lengths at each AWG input port simultaneously, which, in turn,increases the capacity of AWG-based networks significantly.Moreover, by deployingathermalAWGs which do not requireany temperature control, the costs and management of AWG-based networks can be reduced significantly [54].

The wavelength routing properties of the AWG are illustratedin Fig. 1, where six wavelengths [from laser diodes (LDs)] aresent into each input port of an AWG with degree . TheAWG routes every second wavelength to the same output port.This period of the wavelength routing is referred to asfree spec-tral range (FSR). In the depicted example, we exploitFSRs, each consisting of wavelengths. Note that eachFSR provides one wavelength channel for communication be-tween a given AWG input port and a given AWG output port.In addition to the LD wavelengths, a broad-band signal (e.g.,from a LED) which spans all six wavelength channels is sentinto each AWG input port. As illustrated in Fig. 1, the AWGslices the broad-band signal and routes the slices to the respec-tive output ports. Thus, spectral slicing can be used to broadcastlow-speed information, e.g., control information, from a giveninput port to all output ports.

The considered metro WDM network is based on aAWG, as shown in Fig. 2. At each AWG input port, a wave-length-insensitive combiner collects data fromattachednodes. Similarly, at each AWG output port signals are distributedto nodes by a wavelength-insensitive splitter (notethat these splitters can also be used for optical multicasting).Each node is composed of a transmitting part and a receivingpart. The transmitting part of a node is attached to one of thecombiner ports. The receiving part of the same node is located at

Fig. 3. Illustration of spectral spreading of control information.

the opposite splitter port. Thus, the network connectsnodes. Each node contains a tunable LD and a tunable photo-diode (PD) for data transmission and reception, respectively. Inaddition, each node uses a broad-band light source, e.g., LED,for broadcasting control packets by means of spectral slicing.The control information is spread in the electrical domain bymeans of direct sequence spread-spectrum techniques [55], [56].Both data and control signals are combined and then passedthrough the AWG-based network, as illustrated in Fig. 3. At theoutput of the network, a PD is tuned to the same wavelength asthe LD. The PD detects the LD wavelength and correspondingslice of the broad-band signal and converts the combined opticalsignal into the electrical domain. The modulation speed andlaunch power of the broad-band signal are such that the controlsignal has i) a smaller bandwidth, and ii) a smaller power levelthan the data signal [57], as illustrated in the lower right cornerof Fig. 3. Thus, the control signal does not significantly distortthe data signal. The data signal is received without requiringany further processing (except for possibly some high-passfiltering). The control signal is retrieved by low-pass filtering thecombined data and (spread) control signal. The filtered signal isdespread by a decorrelator which multiplies the signal with thecorresponding spreading sequence. The spreading of the control


Fig. 4. Wavelength assignment and timing structure of MAC protocol at a given AWG input port.

signal has the advantage that data and control are sent simulta-neously (i.e., there is no time division between signaling anddata transmission), resulting in increased bandwidth efficiency.

III. MAC PROTOCOL

In this section, we give a brief overview of our MACprotocol; we refer the interested reader to [7] for more details.In our network, each node has a tunable transmitter (TT) anda tunable receiver (TR). This TT-TR structure allows for highflexibility in the data transmissions and receptions and has thepotential to achieve load balancing, as well as improved channelutilization and throughput-delay performance. However, withTT-TR nodes not only channel collisions but also receivercollisions may occur. Typically, a MAC protocol is employedto arbitrate the access to the wavelengths and thus to mitigatecollisions. Generally, MAC protocols for single-hop WDMnetworks fall into the three main categories of i) preallocationprotocols, ii) random-access protocols, and iii) reservation(pretransmission coordination) protocols, comprehensively sur-veyed in [20]. (Since MAC protocols for AWG-based networkshave received little attention so far, the survey [20] studiesthe large body of literature on MAC protocols for PSC-basednetworks. Some key learned lessons from this literature,however, are considered generally valid and guide the design ofthe MAC protocol for our AWG-based network.) Preallocationprotocols statically assign a wavelength to a node during aperiodically recurring time slot. Preallocation generally giveshigh utilization only for uniform nonbursty traffic and is thuspoorly suited for the bursty traffic in future metro networks.Random-access protocols do not require any preallocations tonodes or coordination among nodes. For medium to high traffic

Fig. 5. Frame format.

loads, however, collisions become very frequent resultingin small throughout and large delay. Reservation protocolsemploy pretransmission coordination (reservation signaling)to assign wavelengths and receivers on demand. With theso-called attempt-and-defer type of reservation protocol, datapackets are only transmitted after a successful reservation.Thus, attempt-and-defer protocols completely avoid channeland receiver collisions. This approach is generally preferable ina TT-TR system with bursty traffic and we adopt it for our datapacket transmissions. For the pretransmission coordination wetransmit small control packets according to a random-accessprotocol, namely, a modified slotted ALOHA protocol. Thisapproach is adopted since i) random-access control packettransmission, as opposed to fixed assignments, makes thenetwork scalable, and ii) for the typical large propagation delayto control packet transmission delay ratio, slotted ALOHA issuperior to carrier-sensing based access.

In our MAC protocol, time is divided intocycles, as illustratedin Fig. 4. Each cycle consists of frames. Each frame contains

slots, as illustrated in Fig. 5. The slot length is equal to thetransmission time of a control packet. Each frame is partitioned


into the first , slots and the remainingslots. In the first slots (the shaded area in Figs. 4 and 5), con-trol packet transmissions take place simultaneously with datapacket transmissions which do not exploit spatial wavelengthreuse. In order to avoid receiver collisions of control packets,the receivers at all nodes must be tuned (locked) to one of theLED slices carrying the control information during these first

slots. Owing to the routing characteristics of the AWG, onlynodes attached to the same combiner can send control packets ina given frame. Nodes attached to different AWG input ports sendtheir control packets in different frames. Specifically, all nodesattached to AWG input port (via a commoncombiner) send their control packets in frameof the cycle.Thus, nodes can transmit control packets only in one frame percycle. During the first slots of frame , control and datapackets can be transmitted simultaneously by the nodes attachedto AWG input port . Due to the routing characteristics of theAWG, transmissions from the other AWG input ports cannot bereceived during this time interval. Consequently, the transmit-ters at the other ports cannot send data during this time interval.Thus, wavelengthscannotbe spatially reused during this timeinterval. Therefore, data packets that are longer thanslots can be transmitted only in those frames in which the corre-sponding source node is allowed to send its control packets. Inthe last slots of each frame no control packets are sent.The receivers are unlocked, allowing transmission betweenanypair of nodes ineachframe provided that the corresponding datapacket is not longer than slots. This allows for spa-tial wavelength reuse—the main focus of this paper. Note thatin general, long data packets (length slots) are harderto schedule than short ones (length slots).

When a data packet arrives to a node attached to AWG inputport , the node’s LED broadcasts the corresponding controlpacket in one of the first slots of the frame assigned toAWG input port . The control packet has four fields (see[7]): Destination address (unicast/multicast), length, type(packet/circuit switched), and forward error correction (FEC).The control packet is transmitted on a contention basis using amodified version of slotted ALOHA. Note that using a deriva-tive of slotted ALOHA keeps the control packet contentionsimple, which is of paramount importance in very-high-speednetworks. Furthermore, not assigning each node a dedicatedreservation slot makes the network scalable and allows for newnodes to join the network without service disruptions. Everynode (including the sending node) collects all control packetsby locking its receiver to one of the LED slices carrying thecontrol information during the first slots of every frame.Thus, each node maintains global knowledge of all the othernodes’ activities (and also learns whether its own control packetcollided in the control packet contention or not). Note that thisapproach introduces only a one-way end-to-end propagationdelay (i.e., half the round-trip time). Whereas the conventionalapproach with control packets and explicit acknowledgmentintroduces the two-way end-to-end propagation delay (i.e., thefull round-trip time). Also, note that the control packet broad-cast completely avoids receiver collisions of control packetsand allows each node to learn immediately whether the controlpacket suffered a channel collision. If a control packet collides,

it is retransmitted in the next cycle with probability; withprobability , the retransmission is deferred by one cycle.The successfully received control packets are processed byall nodes in a distributed fashion. Each node applies the samescheduling algorithm and thus comes to the same conclusion.The scheduling algorithm tries to schedule the data packetswithin the scheduling window of frames (i.e., one cycle).If the scheduling fails, the source node retransmits the controlpacket. Given the very-high-speed nature of optical networksand that each node has to process all other nodes’ controlpackets, we employ a simple first-come first-served and first-fitscheduling policy to avoid a computational bottleneck.

IV. A NALYSIS

A. System and Traffic Model

In our analysis, we consider a system with a large. Ouranalysis is approximate for finite and exact in the asymptoticlimit . Throughout our analysis we assume that thepropagation delay is no larger than one cycle (this is reasonablefor a metropolitan area network). Thus, if a control packet issent in a given frame, the corresponding data packet could bescheduled for transmission one cycle later. We assume that allnodes are equidistant from the AWG, i.e., the propagation delayis the same for all nodes (which is easily achieved with low-lossfiber delay lines in real networks).

We assume that each node has a buffer that can hold a singledata packet and a single control packet. We make the followingassumptions about the traffic-generation process. Supposethat a node’s control packet has just been 1) successfullytransmitted, and 2) the corresponding data packet has beensuccessfully scheduled (within the scheduling window of onecycle; see Section IV-C). With probability this node thengenerates the control packet for the next data packet rightbefore the beginning of the next frame in which the node cansend the next control packet (i.e., one cycle after the previouscontrol packet was sent). If no control packet is generated, thenthe node waits for one cycle and then generates a new controlpacket with probability , and so on. The node’s buffer mayhold the scheduled (but not yet transmitted) data packet andthe next control packet at the same time. This next controlpacket is sent with probability one in the next frame assigned tothe node’s AWG input port (possibly simultaneously with thescheduled data packet). A data packet is purged from the node’sbuffer at the end of the frame during which it is transmitted.After a data packet is purged from the buffer, the next datapacket is placed in the buffer, provided the correspondingcontrol packet is already in the buffer.

If a control packet fails in the slotted ALOHA contention orthe data packet scheduling, then the node retransmits the con-trol packet in the next frame assigned to the node’s AWG inputport with probability , with probability it defers the re-transmission by one cycle. In this next cycle, the node transmitsthe control packet with probabilityand defers the transmissionwith probability , and so on. We define

and (1)


We conduct an approximate analysis for large. Our analysisbecomes asymptotically exact when and as well as

(and also ) are fixed [with and chosen so as to satisfy(1)].

We consider uniform unicast traffic. A data packet is destinedto any one of the nodes (including the sending node, for sim-plicity) with equal probability . Let denote the length of adata packet in slots. A data packet is long (has size ) withprobability , i.e., . A data packet is short (hassize ) with probability , i.e.,

. Our model can be extended to more com-plex packet size distributions at the expense of introducing morenotation and a more complex arbitration policy in Section IV-C.

If a control packet fails (either in the slotted ALOHA or thescheduling), the size of the corresponding data packet is notchanged. However, we do assume nonpersistency for the des-tination in our analysis; i.e., a new random destination is drawnfor each attempt of transmitting a control packet. We note thatour comparisons with simulations with persistent destinations(see Section V) clearly show that the analysis provides very ac-curate results despite the destination node nonpersistency as-sumption. We also note that our analysis can be extended to per-sistent destinations in a straightforward manner by maintainingthe random variables introduced in the following for each AWGoutput port.

Now consider the nodes attached to a given (fixed) AWGinput port . These nodes send their controlpackets in frame of a given cycle. We refer to the nodes thatat the beginning of frame hold an old packet, that is, a con-trol packet that has failed in slotted ALOHA or scheduling, as“old.” We refer to all the other nodes as“new.” Note that the setof “new” nodes comprises both the nodes that have generateda new (never before transmitted) control packet as well as thenodes that have deferred the generation of a new control packet.Let be a random variable denoting the number of “new” nodesat AWG input port , and let

(2)

Let be a random variable denoting the number of nodes atport that are to send a control packet corresponding to a longdata packet next (irrespective of whether a given node is “old”or “new,” and keeping in mind that the set of “new” nodesalso comprises those nodes that have deferred the generationof the next control packet; those nodes are accounted for inif the next generated control packet corresponds to a long datapacket). Let be a random variable denoting the number ofnodes at port that are to send a control packet correspondingto a short data packet next. By definition, . Let

(3)

denote the expected fraction of long packets to be sent. We ex-pect that is typically larger than since long packets are harderto schedule and, thus, typically require more retransmissions (ofcontrol packets).

B. Analysis of Control Packet Contention

First, we calculate the number of control packets from nodesattached to AWG input port that are successfulin the slotted ALOHA contention in frame. Let

be a random variable denoting the number of controlpackets that were randomly transmitted in slotby “new” nodes. Recall that each of the“new” nodes sends acontrol packet with probability in the frame. Thus

(4)

Throughout our analysis, we assume thatis large and thatand are fixed. We may, therefore, reasonably approximate theBIN distribution with a Poisson distribution,that is

(5)

which is exact for with fixed. (A refined anal-ysis that does not approximate the binomial distribution by thePoisson distribution is given in Appendix A.) We now recall thedefinition . We also approximate by its expec-tation ; this is reasonable since has only small fluctuationsin steady state for large. Thus

(6)

We note that for , the random variablesare mutually independent. This is because a given node

places with the minuscule probability a control packet in agiven slot, say slot 1. (Note in particular that the expected valueof is small compared to the number of “new” nodes, thatis, ; this is because in the considered asymptotic limit

with fixed, we have .) Thus,has almost no impact on (see Appendix B for aformal proof).

Let be a random variable denoting thenumber of control packets in slot that origi-nate from “old” nodes. Each of the ( ) “old” nodes sends acontrol packet with probability in the frame. Thus

(7)

Approximating this binomial distribution by thePoisson distribution we have

(8)

With and approximating by its expec-tation we get

(9)


We note again that for , the random variablesare mutually independent. They are also indepen-

dent of . Hence, we obtain for

(10)

Henceforth, we let for notational convenience

(11)

i.e.,

(12)

Let , be a random variable indicatingwhether or not slot contains a successful control packet.Specifically, let

if

otherwise.(13)

From (12) clearly,

and

for . The total number of successful controlpackets in the considered frame is , which has a bi-nomial distribution, that is

(14)

Recall from Section IV-A that each packet is destined to anyone of the AWG output ports with equal probability .Let denote the number of successful control packets—in theconsidered frame—that are destined to a given (fixed) AWGoutput port . Clearly, from (14)

(15)

(A refined approximation of which does not ap-proximate the binomial distributions of the ’s and ’s byPoisson distributions is given in Appendix A.) Let denotethe number of successful control packets that correspond to longdata packets destined to a given AWG output port. Recall that

is the expected fraction of long packets to be sent. Hence

BIN

Similarly, let denote the number of control packets that aresuccessful in the slotted ALOHA contention and correspondto short data packets destined to a given AWG output port.Clearly

BIN

C. Analysis of Packet Scheduling

In this subsection, we calculate the expected number ofpackets that are successfully scheduled. Recall from theprevious subsection that the total number of long packets that1) originate from a given AWG input port , 2)

are successful in the slotted ALOHA contention of frame(of a given cycle), and 3) are destined to a given AWG outputport , is BIN . For shortpackets, we have . Note thatthese two random variables are not independent. Let bea random variable denoting the number of long (short) packetsthat 1) originate from a given AWG input port ,2) are successful in the slotted ALOHA contention of frame

(of a given cycle), 3) are destined to a given AWG outputport , and 4) are successfully scheduled within thescheduling window of frames (i.e., one cycle).

Consider the scheduling of packets from a given (fixed) AWGinput port to a given (fixed) AWG output port over the sched-uling window (i.e., frames). Clearly, we can schedule at most

long packets (i.e., ) because the receivers at output portmust tune to the appropriate spectral slices during the first

slots of every frame. Thus, they can tune to a node at AWG inputport for consecutive slots, only in the frame, during whichthe nodes at AWG input portsend their control packets.

Now, suppose that long packets are scheduled (howis determined is discussed shortly). Withlong packets al-

ready scheduled, we can schedule at most

(16)

short packets. To see this, note that in the frame during whichthe nodes at AWG input port send their control packets,there are FSRs—channels between AWG input portand AWG output port —free for a duration of consecutiveslots. Furthermore, there are frames in the schedulingwindow during which the nodes at AWG output portmusttune (are locked) to the nodes sending control packets from theother AWG input ports for the first slots of the frame. Duringeach of these frames, the receivers are unlocked forslots. The utilized FSRs provide parallel channels betweenAWG input port and AWG output port . Note that the

component in (16) is due to thespatial reuse of wavelengths at the considered AWG inputport. Without spatial wavelength reuse this component wouldbe zero and we could schedule at most shortpackets. Continuing our analysis for a network with spatialwavelength reuse, we have

(17)

In (17), we neglect receiver collisions, that is, we do not accountfor situations where a packet cannot be scheduled because its re-ceiver is already scheduled to receive a different packet. This as-sumption is reasonable as receiver collisions are rather unlikelyfor moderately large . This is verified by our simulations whichtake receiver collisions into consideration, see Section V.

In this paper, we consider a first-come first-served first-fitscheduling policy. Data packets are scheduled for the first pos-sible slot(s) at the lowest available wavelength. To arbitrate theaccess to the frame which allows transmission forcontiguousslots and the frames which allow transmission for

contiguous slots we adopt the followingarbitrationpolicy. Our arbitration policy proceeds in one round if there are


or fewer successful control packets in the slotted ALOHAcontention. In case there are more thansuccessful controlpackets in the slotted ALOHA contention, our arbitration policyproceeds in two rounds. First, consider the case whereorfewer control packets are successful in the slotted ALOHA con-tention and we have one round of arbitration. In this case, all thesuccessful packets are scheduled in the frame withavailabletransmission slots. Next, consider the case where more thancontrol packets are successful in the slotted ALOHA contentionand we have two rounds of arbitration. In this case, we scanthe slotted ALOHA slots from index 1 through . In thefirst round, we schedule the first successful packets out ofthe slotted ALOHA contention in the long ( slots) trans-mission slots. In this round, we schedule only one packet foreach of the long transmission slots, irrespective of whether thepacket is long or short. At this point (having filled each of thelong transmission slots with one data packet) all the remainingsuccessful control packets that correspond to long data packetsfail in the scheduling and the transmitting node has to retransmitthe control packet. We then proceed with the second round. Inthe second round, we schedule the remaining successful controlpackets that correspond to short data packets. Provided, we schedule these short data packets for the long transmission

slots that hold only one short data packet from the first round.We also schedule these short data packets for the short (slots) transmission slots. After all the long and short transmis-sion slots have been filled, the remaining short data packets failin the scheduling and the transmitting node has to retransmitthe control packet. We note that our adopted arbitration policyis just one out of many possible arbitration policies, all of whichcan be analyzed in a similar fashion. The first-come first-servedand first-fit scheduling algorithm was chosen to meet the strin-gent timing requirements of reservation-based very-high-speednetworks.

With the adopted arbitration policy the expected number ofscheduled long packets is

(18)

(19)

(20)

To see this, note that in case there are successful controlpackets in the slotted ALOHA contention, then, on average,of these correspond to long data packets. In case , thenthere are on average long packets among the first controlpackets. [If the arbitration policy does not schedule the long (slots) transmission slots first, but, say afterframes that allowtransmission for slots have been scheduled, then anexpected number of long

data packets are scheduled given or moresuccessful control packets in the slotted ALOHA contention.]For notational convenience let

(21)

Thus

(22)

We now calculate the expected number of scheduled shortpackets. Generally

(23)

First, consider the case that there are no more thansuccessfulcontrol packets in the slotted ALOHA contention, i.e., .In this case, we have with (17)

(24)

(25)

since all the successful control packets are scheduled in the longtransmission slots.

Next, consider the case . Let denote thenumber of control packets that correspond to short data packetsto be scheduled in the second round of arbitration. Note that

, because short data packets havebeen scheduled in the first round of arbitration. With (17) weobtain

(26)

(27)

Note that for a network without spatial wavelength reuse, theterm has to be replaced by zero in

(26) and (27), as well as in all the following expressions in thissubsection.

Clearly, , since . More-over, note that conditional on , the random vari-ables and are independent. This is because the firstsuc-cessful control packets in the slotted ALOHA slots determine

; is determined by the subsequent successful con-trol packets. Hence

(28)


(29)

(30)

Now

(31)

(32)

For notational convenience let

(33)

and

(34)

If , then

(35)

(36)

(37)

In case , we have

if

otherwise.

(38)

Combining the cases and , we obtain for, which is typical for practical networks

(39)

(40)

(41)

D. Network/System Analysis

In this subsection, we combine the analyses for the individualcomponents of the considered network, namely, traffic model,slotted ALOHA contention, and packet scheduling. We estab-lish two equilibrium conditions and solve for the two unknowns

and . [Alternatively, we may consider the two unknownsand , noting that for ; the case

is discussed at the end of this section.]In steady state, the system satisfies the equilibrium condition

(42)

To see this, note that in equilibrium the mean number of sched-uled long packets from a given (fixed) AWG input port destinedto a given (fixed) AWG output port [the left-hand side (LHS)] isequal to the mean number of newly generated long packets [theright-hand side (RHS)]. Inserting (22) and (41) in (42) gives

(43)

(44)

The second equilibrium condition is

(45)

This is because new packets are generated in each frame atthe nodes attached to a given AWG input port. With probability

, each of the generated packets is destined to a given (fixed)AWG output port. On the other hand, packets arescheduled (and transmitted) on average from a given AWG inputport to a given AWG output port in one cycle; in equilibrium asmany new packets must be generated. Inserting (1) and (2) inthe LHS of (45) and (22) and (41) in the RHS of (45) we obtain

(46)

Inserting (46) into (44) we obtain

(47)

Inserting (47) into (46) we obtain

(48)

We solve (48) numerically to obtain [noting that by (11),]. We then insert into (47) to

obtain . With and , we calculate (22) and (41).We define the mean throughput as the average number of trans-mitting nodes in steady state (which may also be interpreted asthe average number of successfully transmitted data packets perframe). The mean throughput from a given (fixed) AWG inputport to a given (fixed) AWG output port, i.e., the average numberof nodes transmitting from a given AWG input port to a givenAWG output port in steady state, is then given by

(49)

The mean aggregate throughput of the network is

(50)


Note that and that is bounded by the expression in(16). Thus, if and are integers, the aggregatethroughput is bounded by .

We note that in case , i.e., , we have, from (11),. Inserting this into (44) gives an equation for, which

we solve numerically.We now espouse the mean packet delay in the network. We

define the mean delay as the average time period in cycles fromthe generation of the control packet corresponding to a datapacket until the transmission of the data packet. Recall fromSection IV-C that is the expected number of datapackets that the nodes at a given AWG input port transmit tothe nodes at a given AWG output port per cycle. Now, con-sider a given (fixed) node . In the assumeduniform packet traffic scenario, this node transmits on av-erage data packets to the nodes at a givenAWG output port per cycle. Thus, node transmits on av-erage data packets to the nodes attachedto the AWG output ports per cycle. The average time pe-riod in cycles from the generation of a control packet at node

until the generation of the next control packet is, therefore,. Note that the time period from the

successful scheduling of a data packet until the generation ofthe control packet for the next data packet is geometrically dis-tributed with mean cycles. Hence, the average delayin the network in cycles is

Delay (51)

where and are known from the evaluation of thethroughput (49).

V. NUMERICAL RESULTS

In this section, we show the benefit of spatial wave-length reuse and the impact of the system parameters on thethroughput-delay performance of the network. Data packetscan have one of two lengths. A data packet isslots longwith probability and slots long with probability

. We consider the system parameters number of usedFSRs , the fraction of long data packets, the number ofreservation slots per frame , the physical degree of the AWG

, the number of nodes , and retransmission probability. By default, the parameters take on the following values:

, , , , , , ,and . (For these default parameters the aggregatethroughput is bounded by .) Each cycle is assumed to havea constant length of slots. All numerical resultsin this section are obtained using the expression (15), whichapproximates the number of successful control packets in theslotted ALOHA contention by a Poisson distribution. [For anumerical evaluation of the refined approximation, that doesnot use the Poisson distribution, but uses directly the binomialdistribution, we refer the interested reader to Appendix A.In summary, we find that using the approximate expression(15) gives very accurate results for a wide range of parametervalues, as is also demonstrated by the numerical results inthis section.] We also provide extensive simulation results ofa more realistic network in order to verify the accuracy of the

Fig. 6. Mean delay (cycles) versus mean aggregate throughput (packets perframe) for different number of used FSRsR 2 f1; 2; 3g.

analysis. As opposed to the analysis, in the simulation a givennode cannot transmit data packets to itself and both lengthand destination of a given data packet are not renewed whenretransmitting the corresponding control packet. In addition,the simulation takes receiver collisions into account, i.e., agiven data packet is not scheduled if the receiver of the intendeddestination node is busy. Each simulation was run for 10slotsincluding a warmup phase of 10slots. Using the method ofbatch means we calculated the 98% confidence intervals for themean aggregate throughput and the mean delay whose widthswere less than 1% of the corresponding sample means for allsimulation results.

Fig. 6 shows the mean delay versus the mean aggregatethroughput as the mean arrival rateis varied from to . Asone would expect, using two FSRs instead of one (leaving allother parameters unchanged) dramatically increases the meanaggregate throughput while decreasing the mean delay. This isdue to the fact that an additional FSR increases the degree ofconcurrency and thereby mitigates the scheduling bottleneckresulting in more successfully transmitted data packets andfewer retransmissions. Note that the number of used FSRs islimited and is determined by the transceiver tuning range, thedegree of the underlying AWG, and the channel spacing.To avoid tuning penalties we deploy fast tunable transceiverswhose tuning range is typically 10–15 nm. All results presentedin this section assume a channel spacing of 200 GHz, i.e.,1.6 nm at 1.55 m. Thus, we can use seven to ten wavelengthsat each AWG input port depending on the transceiver tuningrange. For all subsequent results, the number of wavelengths isset to eight. Consequently, with a AWG we deploy twoFSRs for concurrent transmission/reception of data packets.

Figs. 7 and 8 illustrate that spatial wavelength reuse dra-matically improves the throughput-delay performance of thenetwork for variable-size data packets. Fig. 7 shows the meanaggregate throughput versus the mean arrival ratewith andwithout spatial wavelength reuse for different fraction of longdata packets . Simulation and analysis resultsmatch very well. For , i.e., all data packets have a lengthof slots, the mean aggregate throughput is the same no matterwhether wavelengths are spatially reused or not. This is becausethe data packets are too long to be scheduled in the


Fig. 7. Mean aggregate throughput (packets per frame) versus mean arrivalrate� with and without wavelength reuse for different fraction of long datapacketsq 2 f0; 0:5; 1:0g.

Fig. 8. Mean delay (cycles) versus mean arrival rate� with and withoutwavelength reuse for different fraction of long data packetsq 2 f0; 0:5; 1:0g.

frames in which the corresponding nodes do not send controlpackets and spatial wavelength reuse would be possible in thelast slots of the frame. Thus, these frames remainunused for . For , 50% of the data packets arelong ( slots) and the other 50% are short (slots). Allowing forspatial wavelength reuse, the latter ones can now be scheduledin all frames including the aforementioned frames.Consequently, with wavelength reuse, more data packets aresuccessfully transmitted, resulting in a higher throughput. Incontrast, without wavelength reuse, data packets can be sched-uled only in one frame per cycle in which the correspondingnodes also transmit their control packets. Furthermore, since for

some successfully transmitted data packets are short( slots), wavelengths are not fully utilized resulting in a lowerthroughput compared to . For , the benefit ofspatial wavelength reuse becomes even more dramatic. In thiscase, there are only short data packets (slots) which fill upa large number of frames leading to a further increased meanaggregate throughput. Note that for , spatial wavelengthreuse significantly increases the maximum aggregate throughputby more than 60%. All curves in Fig. 7 run into saturation sincefor increasing no additional data packets can be scheduled dueto busy channels and receivers and an increasing number ofcolliding control packets.

Fig. 8 depicts the mean delay versuswith and withoutwavelength reuse for different fraction of long data packets

. We observe that the simulation gives slightlylarger delays than the analysis. This is because the simulationtakes also the transmission time of data packets into accountas opposed to the analysis. In the analysis, the mean delay isequal to the time interval between the generation of a given datapacket and the end of the cycle in which the given data packet issuccessfully scheduled but not yet transmitted. All curves havein common that at very light traffic the mean delay is equal to onecycle owing to the propagation delay of the control packet. Withincreasing the mean delay increases due to more unsuccessfulcontrol packets. These control packets have to be retransmitted,resulting in an increased mean delay. Note that we obtain thelargest delay if the aforementioned frames per cyclecannot be used for data transmission. This holds true not onlyfor the cases where wavelength reuse is not deployed but alsofor with spatial wavelength reuse. This is due to the factthat for the data packets are too long and do not fit inthe last slots of the aforementioned frames.As a consequence, for these cases, fewer data packets can besuccessfully scheduled and the corresponding control packetshave to be retransmitted more often, leading to a higher meandelay. With decreasingthere are more short data packets whichcan easily be scheduled in the aforementioned frames.Due to the resulting wavelength reuse more data packets can besuccessfully scheduled. Therefore, fewer control packets have tobe retransmitted leading to a decreased mean delay. In particular,for , wavelengths are used very efficiently resulting in thelowest mean delay.

The impact of the number of reservation slotsper frame onthe network throughput-delay performance is shown in Figs. 9and 10. The mean aggregate throughput and the mean delay aredepicted as a function of for . Recallthat by default the frame length is set to 200 slots. Each frameis composed of reservation slots and slotswhich can be used for transmitting short packets by means ofspatial wavelength reuse. Clearly, for a fixed, increasingdecreases the length of short packets, and reduces the con-tribution of the short packets to the throughput. At the sametime, increasing increases the probability of successful con-trol packet contention. We observe from Figs. 9 and 10 thatthe effect of increasing the probability of successful controlpacketcontention dominates in the considered scenario, that is,the mean throughput increases and the mean delay decreases as

increases. This indicates that the random-access reservationscheme can be a bottleneck. Indeed, for a very small number of

control slots, we observe the typical bistable behaviorof slotted ALOHA—the underlying mechanism of our controlpacket contention—in the throughput results. Note that the net-work throughput-delay performance could be easily improvedby replacing the random access of the reservation slots with adedicated assignment of the reservation slots. However, such adedicated slot assignment requires reconfiguration when addingnew nodes.

A given number of nodes can be connected by AWGs withdifferent physical degree . Figs. 11 and 12 depict for

the mean aggregate throughput and the mean delay


Fig. 9. Mean aggregate throughput (packets per frame) versus mean arrivalrate� for different number of reservation slotsM 2 f15; 20; 30; 40g.

Fig. 10. Mean delay (cycles) versus mean arrival rate� for different numberof reservation slotsM 2 f15; 20; 30; 40g.

as a function of , respectively. Recall that we have chosen thetransceiver tuning range and the channel spacing such that wemake use of eight wavelengths. The number of used FSRsis then determined only by the physical degreeof the un-derlying AWG and is given by . Consequently, for asmaller , more FSRs are exploited, andvice versafor a larger

. Furthermore, for a smaller each cycle contains fewer butlonger frames, andvice versafor a larger .

As shown in Fig. 11, provides the largest maximummean aggregate throughput at light traffic. However, with in-creasing the mean aggregate throughput decreases. This is dueto the fact that for , short data packets are rather long( slots) resulting in a higher channelutilization and thereby a higher throughput at small traffic loads.But a small also implies that for a given population, morenodes are attached to the same combiner since . Allthese nodes make their reservations in the same frame. For anincreasing , this leads to more collisions of control packets re-sulting in a lower mean aggregate throughput and a higher meandelay due to more retransmissions of the corresponding controlpackets (Fig. 12).

Fig. 11. Mean aggregate throughput (packets per frame) versus mean arrivalrate� for different AWG degreeD 2 f2; 4; 8g.

Fig. 12. Mean delay (cycles) versus mean arrival rate� for different AWGdegreeD 2 f2; 4; 8g.

This problem is alleviated by deploying a or AWG.For a larger , fewer nodes send control packets in the sameframe causing fewer collisions at high traffic loads. However,for and only two FSRs and one FSR can be de-ployed, respectively. Moreover, a larger reduces the lengthof short data packets. Fig. 11 shows that for themean aggregate throughput is rather high for a wide range of

. Whereas for the throughput is rather low due to thesmall number of control packets per frame and the low channelutilization owing to the reduced length of short data packets.Note that for the mean aggregate throughput graduallydecreases for increasing. This is because at high traffic loadscontrol packets suffer from collisions and have to be retrans-mitted, resulting in a slightly higher mean delay compared to

. Concluding, in terms of throughput-delay performancechoosing seems to provide the best solution for a widerange of traffic loads.

Figs. 13 and 14 depict the throughput-delay performance ofthe network for different population .As shown in Fig. 13, increasing improves the mean aggregatethroughput due to more reservation requests and successfullyscheduled data packets. However, for and especially

the throughput decreases for increasing. This


Fig. 13. Mean aggregate throughput (packets per frame) versus mean arrivalrate� for different populationN 2 f40; 100; 200; 300g.

Fig. 14. Mean delay (cycles) versus mean arrival rate� for differentpopulationN 2 f40; 100; 200; 300g.

is because for large populations, more control packets sufferfrom channel collisions resulting in a lower mean aggregatethroughput. Accordingly, this leads to higher mean delays asshown in Fig. 14. Note that simulation and analysis resultsmatch very well even for small populations despite the fact thati) we have conducted an asymptotic analysis for large, andii) the analysis does not take receiver collisions into account(while the simulation does). Also, the analysis assumes non-persistent destinations, whereas the destinations are persistentin the simulation.

The impact of different AWG degree on thesystem throughput-delay performance for different packet sizedistributions is shown in Figs. 15 and 16, respectively. Note thatthroughput and delay are not given as a function ofbut as afunction of the fraction of long data packets . Recallthat we have assumed a constant cycle length of 800 slots and afixed number of reservation slots . As a consequence,the frame length is given by slots and the length ofshort data packets is equal to slots.Moreover, the number of used FSRs of the underlying AWG isgiven by .

Fig. 15 depicts the mean aggregate throughput versus.For , we observe that the throughput monotonously

Fig. 15. Mean aggregate throughput (packets per frame) versus fraction oflong data packetsq for different AWG degreeD 2 f2; 4; 8g.

Fig. 16. Mean delay (cycles) versus fraction of long data packetsq for differentAWG degreeD 2 f2; 4; 8g.

decreases for increasing. For , all data packets are shortand can be scheduled in any frame resulting in a high meanaggregate throughput. For increasing, more and more datapackets are long ( implies that there are only long datapackets). How-ever, long data packets can be scheduled onlyin one frame per cycle. In addition, at most two of them canbe scheduled since . Consequently, for increasing,fewer data packets can be scheduled resulting in a decreasingmean aggregate throughput (Fig. 15) and a higher mean delayas shown in Fig. 16. For , the mean aggregatethroughput is smaller than for and, more interestingly,almost independent from. For , twice as many nodesare attached to each AWG input port compared to . Asa consequence, more control packets suffer from collisionsand fewer data packets are available for scheduling, resultingin a smaller mean aggregate throughput. Moreover, thereare not enough control packets to fully capitalize on spatialwavelength reuse. Thus, for , only slightly more datapackets are successfully scheduled than for . However,since for all successfully scheduled data packets arelong as opposed to , the mean aggregate throughputis about the same in both cases. Similarly, since for


Fig. 17. Mean aggregate throughput (packets per frame) versus mean arrivalrate� for different retransmission probabilityp 2 f0:3; 0:6; 0:9g.

Fig. 18. Mean delay (cycles) versus mean arrival rate� for differentretransmission probabilityp 2 f0:3; 0:6; 0:9g.

fewer nodes are attached to each AWG input port there arefewer reservation requests per frame than for resultingin a smaller throughput. However, these reservation requestsexperience fewer collisions significantly decreasing the meandelay, as illustrated in Fig. 16. In contrast, providesthe highest mean delay due to the large number of collidedcontrol packets and their retransmissions. Note that for all

, the mean delay grows with increasingsincefor larger fewer data packets can be scheduled owing to thelack of spatial wavelength reuse. This leads to more retrans-missions of control packets and thereby to an increased meandelay. Concluding, while suffers from a relativelysmall throughput and exhibits a too large mean delay,choosing seems to provide the best compromise interms of throughput-delay performance.

Figs. 17 and 18 depict the throughput-delay performance ofthe network as a function of for different retransmission prob-ability . As shown in Fig. 17, for

the mean aggregate throughput grows monotonously forincreasing . We observe that at high traffic loads, the slopedecreases due to increasingly busy channels and transceivers.For , the mean aggregate throughput is larger thanfor . This is because with a larger, nodes retransmit

collided control packets with a higher probability resulting inmore successful control packets and an increased mean aggre-gate throughput. However, further increasinghas a detrimentalimpact on the throughput. For , nodes retransmit col-lided control packets after a small backoff period. As a conse-quence, at medium-to-high traffic loads, an increasing numberof control packets collide leading to a decreased mean aggre-gate throughput. Fig. 18 shows that for all ,yields larger mean delays than . With a smaller , nodesdefer retransmissions of collided control packets for a largertime interval which, in turn, increases the mean delay. Notethat for , nodes experience the smallest mean delay atlight-to-medium traffic loads. At high loads, the mean delay be-comes largest due to the increasing number of retransmissionsof control packets.

VI. SUPPLEMENTARY SIMULATION RESULTS

In this section, we give an overview of extensive supplemen-tary simulations of our AWG-based network. In these simula-tions, we examine important additional aspects and performancemetrics of the AWG-based network. We also relax some of thesimplifying assumptions made in the previous analysis in orderto make our investigations more realistic. Specifically, we ex-amine the backoff of our MAC protocol, extend the schedulingwindow size, and equip each node with a finite buffer. In doingso, we relax the assumptions of a one-cycle scheduling windowand a single-packet buffer made in our analytical model. We alsoconduct a benchmark comparison with a previously reportedreservation protocol designed for a PSC-based single-hop metroWDM network. Due to space constraints, we give here only anoverview of these investigations and refer to [58] for more de-tails.

In the simulations, the network parameters take on the fol-lowing default values: , , , ,

, , and . By default, the size of thescheduling window is one cycle and each node’s single-packetbuffer is able to store either a long or a short data packet. Eachcycle has a constant length of slots. The simula-tions are conducted as described in Section V, see also [58] formore details. Throughout this section we report the 95% confi-dence intervals for the performance metrics.

A. Media Access Control (MAC) Protocol Backoff

Without backoff, the retransmission probabilityremainsconstant irrespective of how many times a given control packethas already been retransmitted. As the mean arrival rate in-creases, more nodes are backlogged and try to make a reser-vation in their assigned frame. The network becomes increas-ingly congested and more nodes have to retransmit their unsuc-cessful control packets (with constant probability in ourcase). This leads to an increased number of control packet col-lisions and retransmissions. As a result, for an increasing meanarrival rate, the mean aggregate throughput decreases while themean delay increases dramatically, as depicted in Figs. 19 and20, respectively.

By deploying backoff, this performance degradation is allevi-ated. With backoff, the retransmission probabilityis reduced


Fig. 19. Mean aggregate throughput (packets per frame) versus mean arrivalrate for different backoff limitsb 2 f1; 2; 4; 8g.

Fig. 20. Mean delay (cycles) versus mean arrival rate for different backofflimits b 2 f1; 2; 4; 8g.

each time the reservation fails. More precisely, a given controlpacket which for the first time fails in making a successful reser-vation is retransmitted with the original retransmission proba-bility in the next cycle. If the reservation fails again, then

is reduced by 50%. Thus, the corresponding control packet isretransmitted in the next cycle with probability . Withprobability , the reservation is deferred by onecycle; in that next cycle, the control packet is retransmitted withprobability and deferred with probability ,and so on. Each time the reservation fails,is cut in half. Ingeneral, is reduced by 50% at most times, wheredenotes the backoff limit. Oncehas been halvedtimes, thecontrol packet is retransmitted with probability (where

denotes the original retransmission probability of one inour case) in each of the following cycles until the reservation issuccessful, i.e., there is no limit on the number of attempts.

Figs. 19 and 20 show the positive impact of backoffon the throughput-delay performance of the network for

. We observe that already with , i.e.,

Fig. 21. Mean aggregate throughput (packets per frame) versus mean arrivalrate forb = 4, M = 60, R = 8, and different scheduling window lengthsW 2 f2; 4; 6; 8g frames.

is halved only once, the mean aggregate throughput is sig-nificantly increased for medium-to-high traffic loads, whilethe mean delay is considerably decreased. We observe fromFig. 19 that with , the mean aggregate throughput does notdecrease for an increasing mean arrival rate. This property ofMAC protocols is known asself-stability. Further increasingdoes not yield a better throughput-delay performance. There-fore, we set in the subsequent simulations.

In further simulations that we cannot include here due tospace constraints, we have examined the setting of the numberof reservation slots per frame and the number of used FSRs

for the MAC protocol with backoff, see [58] for details. Inbrief, we find that the mean delay is reduced for larger. Thethroughput, on the other hand, first increases for increasing,reaches a maximum around , and then drops off as isincreased further. We, therefore, set in the subsequentsimulations. We also found that increasing the number of usedFSRs increases the throughput-delay performance signifi-cantly. However, for a given transceiver tuning rangeand AWG degree , a larger results in a smaller channelspacing. Generally, AWGs with a smaller channel spacing ex-hibit a larger crosstalk. In order to achieve acceptable crosstalkvalues we set in the subsequent simulations. Forand a typical fast transceiver tuning range of 12 nm, setting

translates into a channel spacing of 100 GHz.

B. Scheduling Window and Nodal Buffer

Fig. 21 depicts the mean aggregate throughput as a function ofthe mean arrival rate for different scheduling window size

, where is given in frames. We observe that thethroughput is increased when the scheduling window is enlargedfrom two frames (i.e., one cycle for ) to four frames (i.e.,two cycles for ). Further increasing has no significantimpact, as the network resources are almost fully utilized for

. Similar observations hold for the mean delay which isdecreased for larger , see [58]. For the subsequent simulationswe set . This is because in the following we will alsoconsider the case ; in order to benefit from a scheduling


Fig. 22. Relative packet loss versus mean arrival rate forb = 4, M = 60,R = 8,W = 8, and different buffer sizesB 2 f1; 2; 5; 10; 50g packets.

Fig. 23. Mean aggregate throughput (packets per frame) versus mean arrivalrate for b = 4, M = 60, R = 8, W = 8, and different buffer sizes ofB 2 f1; 2; 5; 10; 50g packets.

window of two cycles, the parameter must be equal to eightfor .

Next, we increase the buffer at each node, i.e., thesingle-packet buffer is replaced with a buffer that holds upto long data packets, where . (The low-complexityfirst-in-first-out (FIFO) queuing discipline is considered forour very-high-speed network.) In addition to mean aggregatethroughput and mean delay, we consider the relative packetloss (defined as the ratio of dropped packets to the numberof generated packets). Fig. 22 illustrates the positive impactof larger buffer sizes on the packet loss. With increasing,more arriving data packets can be stored and do not have to bedropped resulting in a decreased packet loss. In addition, eachnode is less likely to be idle which leads to an increased meanaggregate throughput, as depicted in Fig. 23. However, Fig. 24shows that the mean delay significantly increases for larger.

Fig. 24. Mean delay (cycles) versus mean arrival rate forb = 4, M = 60,R = 8,W = 8, and different buffer sizes ofB 2 f1; 2; 5; 10; 50g packets.

Fig. 25. Mean aggregate throughput (packets per frame) forb = 4, M =

60,W = 8, andD 2 f2; 4g, compared to maximum aggregate throughput(packets per frame) of DT-WDMA versus mean arrival rate.

This is because in larger buffers there are packets which haveto wait for a longer time interval until they are transmitted.Clearly, there is a tradeoff between packet loss and delay. Toavoid large delays and provide a reasonable throughput-lossperformance we set in the following simulations.

We next examine the network performance if thenodes are connected by a AWG instead of a AWG.Note that changing the AWG degreeimplies also a different

, , and . For a given transceiver tuning range, a largertranslates into a smaller since . Moreover, we

have considered a fixed cycle length of slots.For we get slots compared toslots for . As a consequence, for , short datapackets are slots and long data packetsare slots long, compared to andslots for , respectively. This results in a decreased meanaggregate throughput, as depicted in Fig. 25 [ignore the dynamictime-wavelength division multiple access (DT-WDMA) curvein this figure for now]. Intuitively, we expect to providea smaller mean delay than since for each cyclecontains twice as many reservation slots as for and the


Fig. 26. Mean delay (cycles) versus mean arrival rate forb = 4, M = 60,W = 8, and differentD 2 f2; 4g.

Fig. 27. Relative packet loss versus mean arrival rate forb = 4, M = 60,W = 8, and differentD 2 f2; 4g.

spatial reuse is doubled. We observe from Fig. 26 that this is trueonly for light-to-medium traffic loads. For a mean arrival ratelarger than about we observe that provides smallerdelays. This is due to the fact that each buffer can store up toten long data packets for whereas for each bufferis able to store up to 20 long data packets owing to the halvedlength of long data packets. As a consequence, for atmedium-to-high loads there are packets which have to wait fora longer time interval until they are transmitted, resulting in anincreased mean delay (and a decreased packet loss, as shown inFig. 27).

C. Comparison With PSC-Based Network

In this section, we compare our AWG-based network andMAC protocol with a PSC-based single-hop metro WDM net-work running the so-calleddynamic time-wavelength divisionmultiple access(DT-WDMA) protocol for resolving packet col-lisions [8]. We have chosen DT-WDMA since among the MACprotocols specifically designed for multiwavelength single-hopWDM networks based on a PSC, DT-WDMA has probably beenthe most influential protocol [59]. Moreover, DT-WDMA has

the following properties in common with our protocol, whichallow for a reasonably fair comparison.

• For data transmission/reception each node is equipped withone singletransceiver.

• Each node’s receiver istunable. Consequently, receiver col-lisions can potentially occur and have to be resolved by the ac-cess protocol.

• DT-WDMA belongs to the category ofreservationproto-cols.

• Resources are dynamically on-demand assigned on aper-packetbasis.

• Nodes are able to acquire and maintainglobal knowledge.• Explicit acknowledgments arenot required.Next, we briefly describe the network architecture and

DT-WDMA (for more details the interested reader is referredto [8]). DT-WDMA is proposed for a metropolitan-sizedsingle-hop WDM network. Each node is equipped with onetransceiver fixed tuned to a common control channel. For dataeach node deploys one transmitter fixed-tuned to a separatewavelength, i.e., each node has its own home channel for trans-mission, and one tunable receiver. Control information is sentover a dedicated signaling channel. Time is divided into slots(which correspond to frames in our protocol) on each channel,and slots on the control channel are further split into minislots(which correspond to slots in our protocol). Fixed time-divisionmultiple access is used within each slot on the control channel,where one minislot is dedicated to each node. Receivers listento the control channel and tune to the appropriate channelto receive packets addressed to them. A common distributedarbitration algorithm is used to resolve conflicts when packetsfrom multiple transmitters contend for the same receiver.

Note that in DT-WDMA each node has its own home channel(wavelength) for transmission, i.e., the number of nodes equalsthe number of wavelengths, while receivers are assumed to betunable over all these wavelengths. Thus, for large populations,receivers with a large tuning range are required whose largetuning time significantly decreases the channel utilization. Inour comparison, we use 16 wavelengths which allows for de-ploying fast tunable receivers in DT-WDMA (and fast tunabletransceivers in our network) whose tuning time is negligible.Furthermore, to compare our above simulation studies withDT-WDMA, we also set the number of nodes to inboth the AWG- and PSC-based networks. To accommodate 200nodes in DT-WDMA we let all data wavelengths be equallyshared among the nodes while we give each of the 200 nodesits own minislot on the control channel. In our comparison wefocus on the throughput. It was shown in [8] that under the sameassumptions on packet arrival process (Bernoulli) and trafficpattern (uniform traffic) as made in our preceding simulations,the maximum utilization of each of the 16 wavelength channelsis in DT-WDMA. This translates into amaximumaggregatethroughput of 9.6 packets/slot (or 9.6 packets/frame,since one slot in DT-WDMA corresponds to one frame in ourprotocol).

Fig. 25 compares the mean aggregate throughput of our net-work versus the mean arrival rate for with the max-imum aggregate throughput of DT-WDMA. We observe thatfor , the mean aggregate throughput of our network


is approximately equal to the maximum aggregate throughputof DT-WDMA at medium-to-high traffic loads. However, for

, our network clearly outperforms DT-WDMA. For awide range of arrival rates, our proposed network provides amean aggregate throughput that is about 70% larger than themaximum aggregate throughput of DT-WDMA. To interpretthis result, recall that 16 wavelengths are deployed in both PSC-and AWG-based networks. However, due to spatial wavelengthreuse, the AWG provides twice as many communicationchannels than the PSC leading to the observed larger aggregatethroughput.

VII. CONCLUSION

We have analyzed the photonic switching of variable-sizepackets with spatial wavelength reuse in an AWG-based metroWDM network. We have obtained computationally efficient andaccurate expressions for the throughput and delay in the net-work. Based on our analytical results, we have conducted ex-tensive numerical investigations of the performance character-istics of the network. We have also conducted extensive simula-tions to verify the accuracy of the analytical results. Our numer-ical results indicate that the AWG-based single-hop network,originally proposed in [7], can efficiently transport packets ofdifferent sizes. We found that spatial wavelength reuse is cru-cial for efficient photonic packet switching. Spatial wavelengthreuse significantly increases the throughput while dramaticallyreducing the delay. We have also demonstrated that the spatialwavelength reuse in the AWG-based network gives a mean ag-gregate throughput that is roughly 70% larger than the max-imum aggregate throughput of a PSC-based network runningthe DT-WDMA protocol.

In our ongoing work we are studying the optimal tradeoffs ofthe network parameters, e.g., the AWG degree that maximizesthe throughput (and minimizes the delay) for a given numberof nodes (and traffic load). We are also studying multicasting inthe AWG-based metro single-hop network.

APPENDIX AREFINED APPROXIMATION OF

In this appendix, we derive a refined approximation for thedistribution of , the number of successful control packets des-tined to a given AWG output port in a given frame. This re-fined approximation does not approximate the binomial distri-butions of the random variables [see (4)] and [see (7)]with Poisson distributions. From (13), we note that

. Recalling that , we have

(52)By the independence of the ’s and ’s we have

(53)Hence, with (4) and (7) we obtain

(54)

TABLE IMEAN AGGREGATETHROUGHPUTTH OBTAINED WITH UNREFINED

APPROXIMATION, REFINED APPROXIMATION, AND SIMULATION FOR

DEFAULT NETWORK PARAMETERS

(55)

(56)

(57)

where (56) follows by approximating by its expectation .Thus, the refined approximation of the distribution ofis givenby

(58)

A. Numerical Evaluation of Refined Approximation for

In this subsection, we evaluate the refined approximationfor and compare it with the approximation (15),which we henceforth refer to as “unrefined.” First, note thatin the unrefined approximation, the binomial distributionBIN is approximated by the Poisson distributionwith parameter [and the BIN distribu-tion is approximated by the Poisson distribution with parameter

]. Clearly, this approximation is accurate when i)is large, ii) is small (and also ), and iii) is large. We

also note that the evaluation of the refined approximation iscomputationally slightly more demanding, as it involves theevaluation of the expression in (56) as compared to the singleexponential term in (15).

In Table I, we compare the mean aggregate throughputobtained with unrefined approximation, refined approximation,and simulation for the typical network parameters chosen as de-fault parameters in Section V [i.e., , , ,

, (and, thus, ), , , and]. We observe that i) the throughput obtained by both

analytical approximations almost coincides, and ii) the analyt-ical results match very well with the simulation results. (Similarobservations hold for the average delay.) In fact, we found inour extensive numerical investigations that these two observa-tions hold for all parameter values considered in Section V. Inparticular, both analytical approximations essentially coincideand match very well with the simulations forvalues as smallas . We also found that both analytical approximations essen-tially coincide and match very well with the simulations forvalues as small as . Table II compares the throughput obtained


TABLE IIMEAN AGGREGATETHROUGHPUTTH OBTAINED WITH UNREFINED

APPROXIMATION, REFINED APPROXIMATION, AND SIMULATION FOR M = 8

with unrefined approximation, refined approximation, and sim-ulation for and , and all other network pa-rameters at their default values. For this smallvalue, we ob-serve that unrefined approximation and refined approximationgive identical throughput for small values ( ),but differ for larger values. We also observe that the refinedapproximation matches the simulation results very well. In sum-mary, we find that the unrefined approximation gives accurateresults for a wide range of network parameters. The refined ap-proximation is more accurate when is small and is large, atthe expense of a slightly more demanding computation. How-ever, small values typically give poor network performance,as is illustrated in Fig. 9, and may therefore not be desirable inpractice.

APPENDIX BPROOF OFINDEPENDENCE OF ’S AND ’S

Let be a sequence of positive integers satisfyingand . Let

be the number of nodes (among allnodes) that send a controlpacket in slot .

Claim: For all nonnegative integers we have

(59)

i.e., the joint distribution of the random variablesapproaches a product of Poisson distributions with identicalparameters . In particular, are asymptoticallyindependent as .

Proof: Fix nonnegative integers and assumethat . The distribution of the randomvector is multinomial withparameters , where(we assume that , so that ). Therefore

(60)

(61)

Hence

(62)

Now note thatA)

(63)

since .B)

(64)because .

C) Since we have and therefore

(65)

This proves the claim.

ACKNOWLEDGMENT

The authors are grateful to Prof. H. Mittelmann of ArizonaState University for providing them with valuable backgroundon numerical analysis.

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Michael Scheutzow received the Diploma degreein mathematics from the Johann-Wolfgang-GoetheUniversity, Frankfurt/Main, Germany, in 1979and the Ph.D. degree in mathematics (magnacum laude) from the University of Kaiserslautern,Kaiserslautern, Germany, in 1983. He received theHabilitation in Mathematics also from the Universityof Kaiserslautern in 1988.

From 1988 through 1990, he was Project Leader atTecmath GmbH, Kaiserslautern. Since 1990, he hasbeen Professor of Stochastics in the Department of

Mathematics at the Technical University of Berlin, Berlin, Germany. From 1997through 1999, he was Associate Chair of the department. He has visited the Uni-versity of Carbondale, Carbondale, IL; Rutgers–The State University of NewJersey, New Brunswick; University of Rochester, Rochester, NY, Warwick Uni-versity, U.K.; and the MSRI at the University of California at Berkeley.

Martin Maier received the Dipl.-Ing. and theDr.-Ing. degrees (both with distinction) in electricalengineering from the Technical University of Berlin,Berlin, Germany, in 1998 and 2003, respectively.

He is currently a Visiting Scholar at the Massa-chusetts Institute of Technology (MIT), Cambridge.He also participates in the national researchproject TransiNet. His research interests includeswitching/routing techniques, protection, architec-tures, and protocols for optical WDM networks.

Dr. Maier was a recipient of the two-year DeutscheTelekom scholarship from June 1999 through May 2001. He is also a corecip-ient of the best paper award presented at the SPIE Photonics East 2000–TerabitOptical Networking Conference.

Martin Reisslein (A’96–S’97–M’98) received theDipl.-Ing. (FH) degree from the FachhochschuleDieburg, Dieburg, Germany, in 1994 and theM.S.E. degree from the University of Pennsylvania,Philadelphia, in 1996, both in electrical engineering.He received the Ph.D. degree in systems engineeringalso from the University of Pennsylvania in 1998.

He is an Assistant Professor in the Department ofElectrical Engineering at Arizona State University,Tempe. During the academic year 1994-1995, hevisited the University of Pennsylvania as a Fulbright

Scholar. From July 1998 through October 2000, he was a scientist withthe German National Research Center for Information Technology (GMDFOKUS), Berlin, Germany. While in Berlin, he was also teaching courses onperformance evaluation and computer networking at the Technical Universityof Berlin. His research interests are in the areas of Internet Quality of Service,video traffic characterization, wireless networking, and optical networking.He maintains an extensive library of video traces for network performanceevaluation, including frame size traces of MPEG–4 and H.263 encoded videoat http://www.eas.asu.edu/trace.

Dr. Resslein has served on the Technical Program Committees of IEEE IN-FOCOM and IEEE GLOBECOM. He is Editor-in-Chief of theIEEE Commu-nications Surveys and Tutorials. He is corecipient of the Best Paper Award ofthe SPIE Photonics East 2000–Terabit Optical Networking Conference.

Adam Wolisz (M’94–SM’99) received the Dipl.- Ingdegree in control engineering, and the Dr.-Ing. degree(both in computer engineering) as well as the Habili-tation in 1972, 1976, and 1983, respectively, all fromthe Silesian Technical University, Gliwice, Poland.

He was with the Polish Academy of Sciencesfrom 1972 to 1989, where he was leading activitieson networking, and, from 1990 to 1993, with theGerman National Research Center for InformationTechnology (GMD FOKUS), heading activities onquantitative aspects of high-speed networks and

multimedia systems. He is a Chaired Professor of Electrical Engineeringand Computer Science at the Technical University of Berlin (TUB), Berlin,Germany, where he is heading the Telecommunication Network Group and theInstitute of Telecommunication Systems. For the period from 2001 to 2003,he is serving as Dean of the Faculty of Electrical Engineering and ComputerScience at TUB. Simultaneously, he is also a member of the Directorial Boardof the Fraunhofer Institute for Open Communication Systems (FhG Fokus).His research interests are in architectures and protocols of communicationnetworks as well as protocol engineering with impact on performance andQuality of Service aspects. Currently, he is working mainly on mobilemultimedia communication, with special regard to architectural aspects ofnetwork heterogeneity and wireless/mobile Internet access. He has authoredtwo books and authored or coauthored over 100 papers in technical journalsand conference proceedings.

Prof. Wolisz is a Member of ITG.

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Wavelength reuse for efficient packet-switched transport